In "Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg Models", Orlov twice mentions the following criterion for a sheaf $P_1$ to be locally free:

If for all closed points $t:x \hookrightarrow X$ we have $Ext^i(P_1, t_* \mathscr{O}_x)=0$ for all $i>0$, then $P_1$ is a locally free sheaf.

I cannot prove this nor find a reference. My only thought is to use adjunction to make this problem local; that is first work with $Ext^i(t^{-1}(P_1), \mathscr{O}_x)$, and then take a left resolution $P^{\cdot} \xrightarrow{\sim} P_1$. Next I'd try to use a spectral sequence such as $$
E_1^{i, j} =
Ext_\mathcal{O}^j(P^{i}, \mathcal{O}_x)
\Rightarrow
Ext_\mathcal{O}^{i + j}(P^{\cdot}, \mathcal{O}_x)=Ext^{i+j}(P_1, \mathcal{O}_x).
$$
but I'm not sure this spectral sequence is valid (I'm trying to use the 2nd spectral sequence on https://stacks.math.columbia.edu/tag/07A9 but derived in the first factor instead of the second; hopefully this introduces a change in sign).